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BUREAU OF MINES 
INFORMATION CIRCUU\R/1989 

C3iO 
1*1 



A Method for the Calibration 
of Class 2 and Class 4 
Standards of Mass 



By Nabii A. Bibawy 



UNITED STATES DEPARTMENT OF THE INTERIOR 



Mission: As the Nation's principal conservation 
agency, the Department of the Interior has respon- 
sibility for most of our nationally-owned public 
lands and natural and cultural resources. This 
includes fostering wise use of our land and water 
resources, protecting our fish and wildlife, pre- 
serving the environmental and cultural values of 
our national parks and historical places, and pro- 
viding for the enjoyment of life through outdoor 
recreation. The Department assesses our energy 
and mineral resources and works to assure that 
their development is in the best interests of all 
our people. The Department also promotes the 
goals of the Take Pride in America campaign by 
encouraging stewardship and citizen responsibil- 
ity for the public lands and promoting citizen par- 
ticipation in their care. The Department also has 
a major responsibility for American Indian reser- 
vation communities and for people who live in 
Island Territories under U.S. Administration. 



Information Circular] 9230 



A Method for the Calibration 
of Class 2 and Class 4 
Standards of Mass 



By Nabil A. Bibawy 



aq^ 



UNITED STATES DEPARTMENT OF THE INTERIOR 
Manuel Lujan, Jr., Secretary 

BUREAU OF MINES 
T S Ary, Director 



i\0' ™* 




CONTENTS 

Page 

Abstract 1 

Introduction 2 

Acknowledgments 2 

Balance and weights 2 

Equations and designs for weighing comparisons 3 

Calibration procedure 4 

Calibration of 1-kg weights 4 

Calibration within a set of weights 4 

Calibration results 5 

Calibration of 5- to 1-kg, class 2 weights , 6 

Calibration of 1-kg, class 2 weights 6 

Calibration of 500- to 100-g, class 2 weights 7 

Calibration of 50- to 10-g, class 2 weights 7 

Calibration of 5- to 1-g, class 2 weights 8 

Calibration of 5- to 1-kg, class 4 weights 8 

Calibration of 1-kg, class 4 weights 9 

Calibration of 500- to 100-g, class 4 weights 9 

Calibration of 50- to 10-g, class 4 weights 10 

Calibration of 5- to 1-g, class 4 weights 10 

Discussion of results 11 

Conclusions 11 

Appendix A.-Multipliers of the observations for determining the calculated correction values and deviation . . 

values for four equal weights X x to X 4 , NIST design A.1.2 12 

Appendix B. -Multipliers of the observations for determining the calculated correction values and deviation . . 

values for weights X 1 to X 6 , less than 1 kg, NIST design C.10.5.2.2.1.1.1 13 

Appendix C— Multipliers of the observations for determining the calculated correction values and deviation . 

values for the calibration of weights X t to X 6 , larger than 1 kg, NIST design C.10.5.2.2.1.1.1 14 

ILLUSTRATION 

1. Voland model HCE 100G balance, class 2 and class 4 sets of weights 3 

TABLES 

1. National Institute of Standards and Technology report of calibration for class M weights 2 

2. Individual comparisons of class 2 weights with the class M standard 5 

3. Individual comparisons of class 4 weights with the class M standard 5 



UNIT OF MEASURE ABBREVIATIONS USED IN THIS REPORT 


g 


gram mg milligram 


kg 


kilogram pet percent 


L 


liter 



A METHOD FOR THE CALIBRATION OF CLASS 2 
AND CLASS 4 STANDARDS OF MASS 



By Nabil A. Bibawy 1 



ABSTRACT 

The U.S. Bureau of Mines has established a method for the calibration of class 2 and class 4 standards 
of mass using a procedure from designs developed by the National Institute of Standards and Technology 
(NIST). This method produces a weight calibration that is within 1.0 pet of the balance sensitivity. A 
set of four 1-kg weights yielded a standard deviation of 3.25 mg compared to ±2.91 mg balance 
sensitivity. The method employs the use of a high-accuracy balance to measure the difference between 
a set of weights calibrated by NIST and the set to be calibrated. The error of the balance is included 
in the error for each weight calibrated. 



Chemist, Helium Field Operations, U.S. Bureau of Mines, Amarillo, TX. 



INTRODUCTION 



The U.S. Bureau of Mines investigates methods to 
improve helium analysis for the certification of calibration 
standards used in helium production, purification, liquifi- 
cation, transportation, and storage operations. Since the 
ultimate composition of primary standard gas mixtures 
depends upon the accuracy of the balance weights used, 
the weights require accurate calibration. Calibrating the 
weights against existing NIST certified weights provides the 
best solution because the error of the balance is included 
in the calibration. 

The true mass of an object is usually determined on an 
analytical balance by comparing the object to a group of 
weights whose individual masses are accurately known. 
True mass of the balance weight is determined by compar- 
ing it to known standards of mass and applying the correc- 
tion value to make the weight agree with the standards. 

Preparation of primary standard gas mixtures for ana- 
lytical instruments was described by Miller, Carroll, and 
Emerson in 1965. 2 The nominal mass of the balance 
weight is the quantity of weight actually observed on the 



balance from comparison to the weight(s) of a weight set 
and the balance chain scale. The correction value is added 
to or subtracted from the nominal weight to obtain the 
true mass. 

Designs for the calibration of standards of mass were 
discussed by Cameron, Croarkin, and Raybold in 1911? 
This collection of designs for the intercomparisons of sets 
of weights was used as a resource for determining the cor- 
rection values for balance weights used in the preparation 
of primary standard gas mixtures. The designs used are 
A.1.2 for the calibration of four equal weights (appendix 
A), and C.10.5.2.2.1.1.1 for the calibration of weights less 
than 1 kg (appendix B), and larger than 1 kg (appendix C). 
These designs provide for a check standard weight, which 
is treated as an additional unknown weight and is used for 
monitoring the performance of the calibration process. 

Because the weights calibrated in this study are com- 
pared against weights of the same metal, the buoyancy 
corrections are not required. 



ACKNOWLEDGMENTS 



The author thanks Dr. Richard S. Davis, physicist, Mass 
Division, Center for Basic Standards, National Institute of 



Standards and Technology, Gaithersburg, MD, for his 
technical assistance. 



BALANCE AND WEIGHTS 



The Voland model HCE 100G 4 analytical balance is a 
double-pan balance used for the comparison of weights 
against standards (fig. 1). The high precision and large 
capacity of this balance make it suitable for the prepara- 
tion of weighed gaseous mixtures in standard-size cylin- 
ders. An analysis using a double-substitution procedure 
to determine the reproducibility of the Voland balance 
yielded a precision of ±2.91 mg. 

Class 2 and class 4 sets of stainless steel weights with 
individual masses ranging from 1 g to 5 kg were supplied 
with the balance. These sets were calibrated using the 
procedures described in this report. 

The weights used as the standard and check standard 
are of a class M set with 1-g to 1-kg masses of stainless 
steel, 30- to 500-mg masses of tantalum, and 1- to 20-mg 
masses of aluminum. This set was made by Wm. 
Ainsworth and Sons, Inc., and calibrated by NIST 
(table 1). 

2 Miller, J. E., A. J. Carroll, and D. E. Emerson. Preparation of 
Primary Standard Gas Mixtures for Analytical Instruments. BuMines 
RI 6674, 1965, 10 pp. 

3 Cameron, J. M., M. C. Croarkin, and R. C. Raybold. Designs for 
the Calibration of Standards of Mass. NBS Tech Note 952, June 1977, 
64 pp.; NTIS PB 268499. 

Reference to specific products does not imply endorsement by the 
U.S. Bureau of Mines. 



Table 1. -National Institute of Standards and Technology 
report of calibration for class M weights 



Item Mass Correction, mq 



1 kg 

500 g 

300 g 

200 g 

100 g 

50g 

30g 

20g 

10 g 

5g 

3g 

2g 

ig 

500 mg 

300 mg 

200 mg 

100 mg 

50 mg 

30 mg 

20 mg 

10 mg 

5mg 

3mg 

2mg 

1 mg 



Apparent 



True 



Uncertainty, 
mg 



2.285 
1.722 

.923 

.627 
-.3111 

.0665 
-.1964 

.0473 

.0166 
-.0239 
-.0228 
-.0184 

.0044 
-.0052 
-.0063 
-.00886 
-.00425 
-.00197 

.00216 
-.00275 

.00384 
-.00434 

.00424 
-.00204 

.00353 



11.039 
6.292 
3.646 
2.443 
.6357 
.5205 
.0761 
.2290 
.1074 
.0215 
.0045 
-.0002 
.0135 
-.0406 
-.0275 
-.02300 
-.01132 
-.00551 
.00004 
.00327 
.00686 
-.00283 
.00515 
-.00144 
.00384 



0.069 
.040 
.034 
.027 
.0075 
.0056 
.0058 
.0050 
.0064 
.0048 
.0033 
.0024 
.0023 
.0017 
.0011 
.00083 
.00078 
.00055 
.00054 
.00046 
.00059 
.00049 
.00052 
.00045 
.00059 



Volume at 
20° C, cm 3 



126.4716 
63.3962 
38.0233 
25.3489 
12.70656 
6.33720 
3.80229 
2.53488 
1.26744 
.63372 
.38023 
.25349 
.12674 
.030118 
.018070 
.012046 
.006023 
.00301 1 
.001807 
.007408 
.003706 
.001850 
.001113 
.000740 
.000371 




Figure 1.-Voland model HCE 100G balance, class 2 and class 4 sets of weights. 

EQUATIONS AND DESIGNS FOR WEIGHING COMPARISONS 



The equations and methods of comparing weights used 
in this report were performed in accordance with NBS 
Technical Note 952 5 and are called designs. 

To calibrate a set of weights, multiple comparison must 
be made of one weight or set of weights against another. 

Using the first comparison as an example, Y x is the 
weight that must be added to the pan carrying X 2 in order 
to balance X^ If weight must be added to X 1 in order to 
balance X 2 , then that weight with a negative sign becomes 
Y 1} indicating that X t is less than X 2 . 



^Work cited in footnote 3. 



For four equal weights, X (1 10 4) , six comparisons are 
made as follows: 



where X, 
and 



(1 to 4) 



(1 to 6) 



x x 


against 


X 2 = Y x 


x x 


against 


X 3 = Y 2 


x x 


against 


X 4 = Y 3 


x 2 


against 


X 3 = Y 4 


x 2 


against 


X 4 = Y 5 


x 3 


against 


X 4 = Y 6 


) = 


weights to be compared 


I 


observed differences for the 




above comparisons. 



The correction values (B) for each weight (or combi- 
nation of weights) were calculated as follows: 



B 1 = [Y 1 K rxY ^ + Y 2 K /v vV + 



B, 






[Y^ _ Y ,+Y 2 K /V .^ + 



: (X 2 ,Y 2 )- 



+ M (X) R]/D 
+ M (X2) R]/D 



The deviations (Dev) are found in a similar manner to 
the B values, using the coefficients from the same designs. 
The deviations are calculated as follows: 



Dev(l) = [Y x L rz Y >> + Y 2 L^ 



WO 



Dev (2) = [Yt L r7 Y >> + Y 2 L,^ 



'(Z 2 ,Y 2 ) 



>]/D 
•]/D 



B ° = ^ K (X n ,Y 1 ) + ^ K (Xn,Y 2 ) + ' • ' +M (Xn) R]/D(l) 



where B n = the calculated correction value for weight 
X,,, mg, 

Y n = the observed difference in the (n)th 
comparison being made, mg, 

*VY Y1 = parameter value for the design being 
^' J used, 



M 



(XJ 



= restraint value for the design being used, 



and 



R = the previously determined true mass 
correction value for the standard weight 
or combination of standard weights, mg, 

D = the divisor for the design being used. 



The true mass for each weight (X) is determined by 
adding or subtracting the correction value B to the nom- 
inal value of the standard weights being compared. 



Dev (n) 



[Y l L (7 Y ^ + Y 2 L(Z n ,Y 2 ) + 



Wi) 



-]/D(2) 



where Dev (n) = 

Y n = 

L (Z n ,YJ - 

and D = 



calculated deviation corresponding 
to observation Y n , mg, 

the observed difference in the (n)th 
comparison, mg, 

the parameter value (deviation) 
from the design being used, 

the divisor for the design being 
used. 



The standard deviation of the particular set of measure- 
ments is calculated as follows: The deviations are com- 
puted, squared, summed, and divided by the degrees of 
freedom for the particular design being used. The square 
root of this number is the standard deviation for the 
measurements. This should not be significantly different 
from the standard deviation of the balance of ± 2.91 mg. 



CALIBRATION PROCEDURE 



CALIBRATION OF 1-kg WEIGHTS 

NIST design A.1.2 (appendix A) was used in the cali- 
bration procedure for a 1-kg weight or a combination of 
weights equivalent to 1 kg. In the calibration procedure, 
a 1-kg weight (Xj) and a combination of weights equaling 
1 kg (X 2 ) from the desired set were compared to an NIST- 
calibrated class M check standard (X 3 ) and standard (X 4 ). 
The check standard was a combination of weights equaling 
1 kg, and the standard was a 1-kg weight. 

CALIBRATION WITHIN A SET OF WEIGHTS 

NIST design C.10.5.2.2.1.1.1 (appendix B) was used in 
the calibration procedure for weight sets with denomina- 
tions of 500 g, 200 g, 200 g, and 100 g; 50 g, 20 g, 20 g, and 
10 g; and 5 g, 2 g, 2 g, and 1 g; or other combinations with 
the ratio of 5:2:2:1. In the calibration procedure, a 500-g, 
50-g, or 5-g weight was used as X l5 a 200-g, 20-g, or 2-g 
weight was used as X 2 , a 200-g, 20-g, or 2-g weight was 
used as X 3 , and a 100-g, 10-g, or 1-g weight was used as 
X 4 . The desired set was compared to a check standard 
(X 5 ) comprised of a combination of 50-g, 20-g, 20g, and 



10-g weights, 5-g, 2-g, 2-g, and 1-g weights, or 500-mg, 
200-mg, 200-mg, and 100-mg weights. 

Weight X 6 was the standard obtained from the NIST- 
calibrated class M set and was a 100-g, 10-g, or 1-g weight. 
Eight different comparisons were made on the Voland 
balance as follows: 



x 1+ x 6 


against 


X 2 +X 3 +X 4 +X5 = 


Y, 


x 1+ x 5 


against 


X 2 +X 3 + X 4 +X 6 = 


Y 2 


x 1+ x 4 


against 


x 2 +x 3 +x 5 +x 6 = 


Y 3 


x x 


against 


x 2 +x 4 +x 5 +x 6 = 


Y 4 


X t 


against 


x 3 +x 4 +x 5 +x 6 = 


Y 5 


x 2 +x 4 


against 


x 3 +x 5 


Y 6 


X.+X, 


against 


x 3 +x 4 


Y 7 


x 2 +x 5 


against 


Xs + X, 


Y 8 



The Y (i to 8) values are the observations obtained for 
the above comparisons. The correction values (B) and 
deviations were calculated using equations 1 and 2, 
respectively, and the parameters and restraint from NIST 
design C.10.5.2.2.1.1.1 shown in appendix B. The standard 
deviation of a particular set of measurements was calcu- 
lated using the method outlined previously. 



To calibrate weights larger than 1 kg, the same NIST 
design (C.10.5.2.2.1.1.1) was used, but with a different 
restraint, divisor, and parameter values as shown in 



appendix C. Correction values, deviations, and the stand- 
ard deviations were calculated using equations 1 and 2 and 
the same steps as outlined above. 



CALIBRATION RESULTS 



The calibration results of the class 2 and class 4 sets 
using the class M, NIST-calibrated set as standards are 



given in tables 2 and 3 respectively. The results of cali- 
brations for individual weights follow. 



Table 2.-lndividual comparisons of class 2 weights with the class M standard 



Weight 


Item 


Identification 


True mass of, 


Standard 


Standard 






mark 


correction, 


deviation of 


deviation for 








mg 


correction, mg 


a set, mg 


1 


5 kg 


None 


93.052 


7.06 


3.62 


2 


2 kg 


X 


34.935 


3.69 


3.62 


3 


2 kg 


None 


18.507 


3.69 


3.62 


4 


1 kg 


None 


14.789 


2.30 


2.04 


5 


500 g 


None 


9.091 


1.06 


2.67 


6 


200 g 


X 


3.065 


1.25 


2.67 


7 


200 g 


None 


.208 


1.25 


2.67 


8 


100 g 


None 


3.675 


1.41 


2.67 


9 


50 g 


None 


2.909 


1.31 


2.44 


10 


20 g 


X 


-.122 


1.29 


2.44 


11 


20 g 


None 


-1.551 


1.29 


2.44 


12 


10 g 


None 


.296 


1.42 


2.44 


13 


5g 


None 


1.577 


1.31 


2.18 


14 


2g 


None 


2.059 


1.29 


2.18 


15 


2g 


X 


-.798 


1.29 


2.18 


16 


ig 


None 


-.399 


1.42 


2.18 



Table 3.-lndividual comparisons of class 4 weights with the class M standard 



Weight 


Item 


Identification 


True mass of, 


Standard 


Standard 






mark 


correction, 


deviation of 


deviation for 








mg 


correction, mg 


a set, mg 


1 


5 kg 






66.624 


7.06 


5.87 


2 


2 kg 




•X 


25.649 


3.69 


5.87 


3 


2 kg 






15.649 


3.69 


5.87 


4 


1 kg 






16.039 


2.30 


2.89 


5 


500 g 






10.698 


1.06 


1.54 


6 


200 g 




•X 


.279 


1.25 


1.54 


7 


200 g 






.279 


1.25 


1.54 


8 


100 g 






2.283 


1.41 


1.54 


9 


50 g 






4.713 


1.31 


4.42 


10 


20 g 




•X 


.885 


1.29 


4.42 


11 


20 g 






-4.115 


1.29 


4.42 


12 


10 g 






.800 


1.42 


4.42 


13 


5g 






2.543 


1.31 


2.44 


14 


2g 




•X 


.874 


1.29 


2.44 


15 


2g 






.160 


1.29 


2.44 


16 


ig 






-.634 


1.42 


2.44 



CALIBRATION OF 5- TO 1-kg, 
CLASS 2 WEIGHTS 

Calibration of the 5- to 1-kg, class 2 set of weights was 
accomplished using the procedure outlined under 
"Calibration Within a Set of Weights" and NIST design 
C.10.5.2.2.1.1.1 (appendix C). 

Observations 



Y (1) 

Zw 

v (5) 



10 mg 
30 mg 
30 mg 
10 mg 
25 mg 
20 mg 
10 mg 
20 mg 
11.039 mg 



These observation values and R x were then used in 
equation 1 to calculate the true masses, which are 

X 1 = 5 kg + 93.052 mg, 
X 2 = 2 kg + 34.935 mg, 

and X 3 = 2 kg + 18.507 mg, 

The deviations were 



Dev (1 
Dev (2 
Dev (3 
Dev (4 
Dev (5 
Dev (6 
Dev (7 
Dev (8 



-2.86 mg 
2.86 mg 

0.71 mg 

-0.71 mg 
2.14 mg 
2.14 mg 

-3.57 mg 



The sum of the deviations squared is 39.27, which divided 
by 3 degrees of freedom is 13.09. The square root of 13.09 
is 3.62 mg, the standard deviation of this set of measure- 
ments. 

R^s found in the report of calibration for the class M set standard 
for 1-kg weight (table 1). 



CALIBRATION OF 1-kg, 
CLASS 2 WEIGHTS 

Calibration of the 1-kg, class 2 weight was accomplished 
using the procedure outlined under "Calibration of 1-kg 
Weights" and NIST design A.1.2 (appendix A). 



Observations 



Y (1) 

Zw 

v (5) 

3P 







5mg 

5mg 

5mg 



11.039 mg 



These observation values and R t were then used in 
equation 1 to calculate the true masses, which are 



and 



X : = kg + 14.789 mg 
X 2 - kg + 16.039 mg. 



R 2 = 16.039 mg, which will be the restraint used in the 
next weighing set. 



The deviations were 



Dev (1) 
Dev (2) 
Dev (3) 
Dev (4) 
Dev (5) 
Dev (6) 



1.25 mg 
-2.50 mg 

1.25 mg 

1.25 mg 


-1.25 mg 



The sum of the deviations squared is 12.50, which divided 
by 3 degrees of freedom is 4.17. The square root of 4.17 
is 2.04 mg, the standard deviation of this set of measure- 
ments. 



7 Rj is found in the report of calibration for the class M set (table 1) 
and is the true mass correction value for the 1-kg weight. 



CALIBRATION OF 500- TO 100-g, 
CLASS 2 WEIGHTS 

Calibration of the 500- to 100-g, class 2 set of weights 
was accomplished using the procedure outlined under 
"Calibration Within a Set of Weights" and NIST design 
C.10.5.2.2.1.1.1 (appendix B). 

Observations 

lo-) = 5 n mg 

Y( 2 ) = Omg 

Y (3) = 5mg 

Y (4) = Omg 

Y(5) = Omg 

Y (7) = Omg 
Y (8) = 5mg 
8 R 2 - 16.039 mg 

These observation values and R 2 are then used in 
equation 1 to calculate the true masses, which are 

X x = 500 g + 9.091 mg, 

X 2 = 200 g + 3.065 mg, 

X 3 = 200 g + 0.208 mg, 

X 4 = 100 g + 3.675 mg, 

and Xs = 100 g + 1.533 mg. 

R 3 = 1.533 mg, which will be the restraint used in the 
next weighing set. 

The deviations were 



Dev (1) = 


= 2.14 mg 


Dev (2) = 


= -1.43 mg 


Dev (3) = 


= -0.71 mg 


Dev (4) = 


1.43 mg 


Dev (5) = 


= -1.43 mg 


Dev (6) = 


-- 


Dev (7) = 


= -1.43 mg 


Dev (8) = 


= 2.86 mg 



The sum of the deviations squared is 21.44, which divided 
by 3 degrees of freedom is 7.15. The square root of 7.15 
is 2.67 mg, the standard deviation of this set of measure- 
ments. 



CALIBRATION OF 50- TO 10-g, 
CLASS 2 WEIGHTS 

Calibration of the 50- to 10-g, class 2 set of weights was 
accomplished using the procedure outlined under 
"Calibration Within a Set of Weights" and NIST design 
C.10.5.2.2.1.1.1 (appendix B). 

Observations 

Y(i) = 5 mg 

Y(2) = 5 mg 

Y 3 = 

Y 4 = 

Y 5 = 

Y 6 = 
Y (7) = ° 
I( 8 ) = ?S5 



*r; 



1.533 mg 



These observation values and R 3 were then used in 
equation 1 to calculate the true masses, which are 

Xj = 50 g + 2.909 mg, 

X 2 = 20 g + (-0.122) mg, 

X 3 = 20 g + (-1.551) mg, 

X 4 = 10 g + 0.296 mg, 

and Xs = 10 g + 2.439 mg. 

R 4 = 2.439 mg, which will be the restraint used in the 
next weighing set. 



The deviations were 



Dev(l 
Dev (2 
Dev (3 
Dev (4 
Dev (5 
Dev (6 
Dev (7 
Dev (8 



2.14 mg 
-0.71 mg 
-1.43 mg 

0.71 mg 
-0.71 mg 

0.71 mg 
-2.14 mg 

2.14 mg 



The sum of the deviations squared is 17.80, which divided 
by 3 degrees of freedom is 5.93. The square root of 5.93 
is 2.44 mg, the standard deviation of this set of measure- 
ments. 



Rj is found in the calibration of the 1-kg, class 2 weight and is the 



'Rj is found in the calibration data of the 500- to 100-g, class 2 set 



CALIBRATION OF 5- TO 1-g, 
CLASS 2 WEIGHTS 



CALIBRATION OF 5- TO 1-kg, 
CLASS 4 WEIGHTS 



Calibration of the 5- to 1-g, class 2 set of weights was 
accomplished using the procedure outlined under 
"Calibration Within a Set of Weights" and NIST design 
C.10.5.2.2.1.1.1 (appendix B). 

Observations 



Y (1) 

v (5) 
Y (6) 

Y(7) - 5mg 

Tp : 5mg 



= 

= 

= 

= 

= 

= 



Calibration of the 5- to 1-kg, class 4 set of weights was 
accomplished using the procedure outlined under "Calibra- 
tion of 1-kg Weights." Comparisons and calculation of 
correction values, deviations, and the standard devia- 
tion were made using NIST design C.10.5.2.2.1.1.1 
(appendix C). 

Observations 



2.439 mg 



These observation values and R 4 were then used in 
equation 1 to calculate the true masses, which are 



Y (1) 

Zw 

v (5) 

Zf® 
3g> 



20 mg 
15 mg 
10 mg 
15 mg 
15 mg 
10 mg 
10 mg 
15 mg 
11.039 mg 



These observation values and R : were then used in 



and 



The deviations were 



Xj - 5 


g + 


1.577 mg, 


equation 1 to calculate the true masses, which ; 


X 2 = 2 


g + 


2.059 mg, 


X, = 


5 kg 


+ 66.624 mg, 


X 3 = 2 


g + 


(-0.798) mg, 


x 2 = 


2 kg 


+ 25.649 mg, 


X 4 = 1 

were 


g + 


(-0.399) mg. 


and X 3 = 
The deviations were 


2 kg 


+ 15.649 mg. 


Dev (1) 


= 


Omg 


Dev (1) 


= 


3.57 mg 


Dev (2) 


= 


-1.43 mg 


Dev (2) 


= 


-1.43 mg 


Dev (3) 


= 


1.43 mg 


Dev (3) 


= 


-2.14 mg 


Dev (4) 


= 


1.43 mg 


Dev (4) 


= 


5.00 mg 


Dev (5) 


= 


-1.43 mg 


Dev (5) 


= 


-5.00 mg 


Dev (6) 


= 


-1.43 mg 


Dev (6) 


= 


2.14 mg 


Dev (7) 


= 


1.43 mg 


Dev (7) 


= 


-2.14 mg 


Dev (8) 


= 


1.43 mg 


Dev (8) 


= 


5.00 mg 



The sum of the deviations squared is 14.31, which divided 
by 3 degrees of freedom is 4.77. The square root of 4.77 
is 2.18 mg, the standard deviation of this set of measure- 
ments. 



The sum of the deviations squared is 103.53, which divided 
by 3 degrees of freedom is 34.51. The square root of 34.51 
is 5.87 mg, the standard deviation of this set of measure- 
ments. 



10 R 4 is found in the calibration data of the 50- to 10-g, class 2 set 
and is the true mass correction value for mass X 5 . 



Rj is found in the report of calibration for the class M set 
(table 1) and is the true mass correction value for the 1-kg weight. 



CALIBRATION OF 1-kg, 
CLASS 4 WEIGHTS 



CALIBRATION OF 500- TO 100-g, 
CLASS 4 WEIGHTS 



For calibration of the 1-kg weight, the procedure 
outlined in the "Calibration of 1-kg Weights" section was 
used. NIST design A.1.2 (appendix A) was employed for 
calculation of the correction values, deviation values, and 
standard deviation. 

Observations 



Y (1) 
Y (5) 

Y(6) 

12 R, = 



5mg 


5 mg 

5mg 



= 



11.039 mg 



These observation values and R x were then used in 
equation 1 to calculate the true masses, which are 



and 



X, = 1 kg + 16.039 mg, 
X^ = 1 kg + 13.539 mg. 



The deviations were 



Dev (1) 
Dev (2) 
Dev (3) 
Dev (4) 
Dev (5) 
Dev (6) 



2.50 mg 
-2.50 mg 



2.50 mg 
-2.50 mg 



The sum of the deviations squared is 25.0, which divided 
by 3 degrees of freedom is 8.33. The square root of 8.33 
is 2.89 mg, the standard deviation of this set of measure- 
ments. 



Calibration of the 500- to 100-g, class 4 weights was 
accomplished by following the steps outlined under "Cali- 
bration Within a Set of Weights." Comparisons and calcu- 
lation of the correction values, deviations values, and 
standard deviation were made using NIST design 
C.10.5.2.2.1.1.1 (appendix B). 

Observations 



Y (1) 

I® 

v (5) 



*8 : 



5 mg 
10 mg 
10 mg 

5 mg 

5mg 

Omg 

Omg 

Omg 
13.539 mg 



These observation values and R 5 were then used in 
equation 1 to calculate the true masses, which are 

Xj = 500 g + 10.698 mg, 



R 5 = 13.539 mg, which will be the restraint used in the j 

next weighing set. 



X 2 = 

X 3 = 

x 4 = 

X, = 



200 g + 0.279 mg, 

200 g + 0.279 mg, 

100 g + 2.283 mg, 

100 g + 2.283 mg. 



R 6 = 2.283 mg, which will be the restraint used in the 
next weighing set. 

The deviations were 



Dev (1) = 
Dev (2) = 
Dev (3) = 
Dev (4) = 


- -1.43 mg 
= 0.71 mg 
= 0.71 mg 

= 


Dev (5) = 


= 


Dev (6) -- 


= 


Dev (7) = 
Dev (8) = 


= 1.43 mg 
= -1.43 mg 



The sum of the deviations squared is 7.14, which divided 
by 3 degrees of freedom is 2.38. The square root of 2.38 
is 1.54 mg, the standard deviation of this set of measure- 
ments. 



12 Rj is found in the report of calibration for class M set (table 1) ^Rj is found in the report of calibration of the 1-kg, class 4 weight 



and is the true mass correction value for mass X 2 . 



10 



CALIBRATION OF 50-to 10-g, 
CLASS 4 WEIGHTS 

Calibration of the 50- to 10-g, class 4 set of weights was 
accomplished using the procedure outlined under 
"Calibration Within a Set of Weights." The comparison 
and calculation of correction values, deviations, and the 
standard deviation were made using NIST design 
C.10.5.2.2.1.1.1 (appendix B). 

Observations 



Y m 


= 


5mg 


Y (2) 


= 


10 mg 


*0) 


= 


5mg 


Y (4) 


= 


5 mg 


Y (5) 


= 





*w 


= 


5 mg 


Y (7) 


= 


5 mg 


T(8) 


= 


10 mg 


X 


= 


2.283 mg 



These observation values and R 6 were then used in 
equation 1 to calculate the true masses, which are 

Xj = 50 g + 4.713 mg, 

X 2 = 20 g + 0.885 mg, 

X 3 = 20 g + (-4.115) mg, 

X 4 = 10 g + 0.800 mg, 

and X 5 - 10 g + 2.943 mg. 

R 7 = 2.943 mg, which will be the restraint used in the 
next weighing set. 

The deviations were 



Dev (1) = 
Dev (2) = 
Dev (3) = 


= 0.71 mg 
= -1.43 mg 
= -0.71 mg 


Dev (4) = 
Dev (5) = 


= 5.00 mg 
= -5.00 mg 


Dev (6) = 


= 0.71 mg 


Dev (7) = 

Dev (8) = 


- 0.71 mg 
= 2.14 mg 



The sum of the deviations squared is 58.67, which divided 
by 3 degrees of freedom is 19.56. The square root of 19.56 
is 4.42 mg, the standard deviation of this set of 
measurements. 

14 Rg is found in the calibration data of the 500- to 100-g, class 4 



CALIBRATION OF 5- TO 1-g, 
CLASS 4 WEIGHTS 

Calibration of the 5- to 1-g, class 4 set of weights 
employed the procedure outlined under "Calibration 
Within a Set of Weights." The comparisons and calcula- 
tion of the correction values, deviations, and standard 
deviation were made using NIST design C.10.5.2.2.1.1.1 
(appendix B). 

Observations 



Y 



[(.2) 

:( 3 ) 

(4) 

:(8) 



= 5mg 

- 

= 

= 

= 5mg 

= 

= 

= 



15 R 7 - 2.943 mg 

These observation values and R 7 were then used in 
equation 1 to calculate the true masses, which are 

X x - 5 g + 2.543 mg, 

X 2 - 2 g + 0.874 mg, 

X 3 = 2 g + 0.160 mg, 

and X 4 = 1 g + (-0.634) mg. 

The deviations were 



Dev(l 
Dev (2 
Dev (3 
Dev (4 
Dev (5 
Dev (6 
Dev (7 
Dev (8 



1.43 mg 
-0.71 mg 
-0.71 mg 
-2.14 mg 
2.14 mg 
-0.71 mg 
-2.14 mg 
0.71 mg 



The sum of the deviations squared is 17.80, which divided 
by 3 degrees of freedom is 5.93. The square root of 5.93 
is 2.44 mg, the standard deviation of this set of 
measurements. 



15 Rj is found in the calibration data of the 50- to 10-g, class 4 set 
and is the true mass correction value for mass Xj. 



11 



DISCUSSION OF RESULTS 



The data reported in tables 2 and 3 show the precision 
that can be achieved using the Voland model HCE 100G 
analytical balance as a comparator, when calibrating 



class 2 and class 4 sets of weights. The standard deviations 
of each set of weights were generally within ± 1 mg of the 
±2.91-mg balance sensitivity. 



CONCLUSIONS 



This method of calibration of standards of mass is 
adequate to obtain the true masses of weights that are 
used as counterbalances in determining the true masses of 
individual gases in primary gas standard preparation. A 
unique feature of this method is that weighing designs are 
employed that provide a self- consistency check of the 
calibration process. 

The standard deviation for the class 2 weights, when 
compared to the NIST class M calibrated 1-kg weight, was 
determined for each set. The standard deviation was 



±3.62 mg for the 5-kg weight set, ±2.67 mg for the 500-g 
weight set, ±2.44 mg for the 50-g weight set, and ±2.18 
mg for the 1-g weight set. 

The standard deviation for the class 4 weights, when 
compared to the NIST class M calibrated 1-kg weight, was 
determined for each set. The standard deviation was 
± 5.87 mg for the 5-kg weight set, ± 1.54 mg for the 500-g 
weight set, ±4.42 mg for the 50-g weight set, and ±2.44 
mg for the 1-g weight set. 



12 



APPENDIX A.— MULTIPLIERS OF THE OBSERVATIONS FOR DETERMINING 

THE CALCULATED CORRECTION VALUES AND DEVIATION VALUES 

FOR FOUR EQUAL WEIGHTS X 1 TO X 4 , NIST DESIGN A.1.2 

The restraint for this design is that weight x 4 has previously been calibrated and is used for the known standard. 

Parameter values [K(x y )]> * divisor = 4 

Observations X 1 Xj X 3 X 4 

Yj 1 A (T 

Y 2 1 -1 

Y 3 2 1 1 

Y 4 1 -1 

Y 5 1 2 1 

Y 6 1 1 2 

M 4 4 4 4~ 



where 

degrees of freedom 

and 



* Defined in equation 1. 

K-(X y ) = parameter values, 

= 3, 

M^x ) = restraint values for the design. 



Deviation parameter values [L/£ y )]>** divisor = 4 

Observations Z : Z 2 Zj Z 4 Z 5 Z 6 

Yj 2 ^ ^1 1 1 0~ 

Y 2 -1 2 -1 -1 1 

Y 3 -1 -1 2 -1 -1 

Y 4 1 -1 2 -1 1 

Y 5 1 -1 -1 2 -1 

Y 6 1 -1 1 -1 2 



where 



** Defined in equation 2. 
L(Z ,Y ) = deviation values for the design. 



13 



APPENDIX B.— MULTIPLIERS OF THE OBSERVATIONS FOR DETERMINING 

THE CALCULATED CORRECTION VALUES AND DEVIATION VALUES FOR 

WEIGHTS X, TO X 6 , LESS THAN 1 kg, NIST DESIGN C.1 0.5.2.2.1. 1.1 

The restraint for this design is that the sum of weights X 1} X 2 , X 3 , and X 4 has previously been calibrated and is used 
for the known standard. 

Parameter values [Kyy Y )]> * divisor = 70 

Observations X x Xj X 3 X 4 X 5 Xg 

Y 1 15 ^8 ^8 1 1 2T 

Y 2 15 -8 -8 1 21 1 

Y 3 5-12-12 19 -1 -1 

Y 4 2 12-14-14 -14 

Y 5 12 2-14-14 -14 

Y 6 -5 8-12 9-11 -1 

Y 7 5 12 -8 -9 1 11 

Y 8 10-10 10-10 

M 35 14 14 7 7 T 



where 

degrees of freedom 

and 



* Defined in equation 1. 
Kvv Y ) = parameter values, 
= 3, 
M/v -\ = restraint values for the design. 



Deviation parameter values [L/£ y )]>** divisor = 7 

Observations Z x Z 2 Zj Z 4 Z 5 Z 6 Zy 

YJ 2 ^1 ~\ C) 3 

Y 2 -1 2 -1 2 

Y 3 -1 -1 2 -2 2 

Y 4 3 -3 1 1 

Y 5 -3 3 -1 -1 

Y 6 2 -2 1 -1 3 -1 

Y 7 -2 2 1 -1 -1 3 

Y fi 2 -2 1 -1 -1 -1 



where 



** Defined in equation 2. 
L(Z Y ) = deviation values for the design. 



2 
-2 



1 
-1 
-1 
-1 

3 



14 



APPENDIX C.— MULTIPLIERS OF THE OBSERVATIONS FOR DETERMINING THE 

CALCULATED CORRECTION VALUES AND DEVIATIONS FOR THE CALIBRATION 

OF WEIGHTS X, TO X 6 , LARGER THAN 1 kg, NIST DESIGN C.1 0.5.2.2.1. 1.1 

The restraint for this design is that weight ^ has previously been calibrated and is used for the known standard. 

Parameter values [L/^ Y )]» * divisor = 7 

Observations X 1 X 2 X 3 X 4 Xj X$ 

YJ ^9 ^5 ^5 ^2 ^2 0~ 

Y 2 1 -1 -1 2 

Y 3 1 -1 -1 2 

Y 4 7 3 4 

Y 5 7 4 3 

Y 6 1 -1 1 -1 

Y 7 -5 -1 -3 -2 -1 

Y 8 5 3 1 1 2 

M 35 14 14 7 7 7~ 



where 

degrees of freedom 

and 



* Defined in equation 1. 
K/y Y ) = parameter values for the design, 
= 3, 
M/vj = the corresponding restraint values for the design. 



where 



Deviation parameter values [L/^ Y )]»** divisor = 7 

Observations Zj Z 2 Zj Z 4 Z 5 Z 6 Z, 

Y : 2 ^ ^1 3 

Y 2 -1 2 -1 2 

Y 3 -1 -1 2 -2 2 

Y 4 3 -3 1 1 

Y 5 -3 3 -1 -1 

Y 6 2 -2 1 -1 3 -1 

Y 7 -2 2 1 -1 -1 3 

Y 8 2 -2 1 -1 -1 -1 



** Defined in equation 2. 
L(Z Y ) = deviation values for the design. 



2 
-2 



1 
-1 
-1 
-1 

3 



U.S. GOVERNMENT PRINTING OFFICE: 611-012/00,117 



INT.BU.OF MINES,PGH.,PA 28993 



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